Phase Space Electronic Structure Theory: From Diatomic Lambda-Doubling to Macroscopic Einstein–de Haas

Λ-doubling of diatomic molecules is a subtle microscopic phenomenon that has long attracted the attention of experimental groups, insofar as rotation of molecular nuclei induces small energetic changes in the (degenerate) electronic state. A direct description of such a phenomenon clearly requires going beyond the Born–Oppenheimer approximation. Here we show that a phase space theory previously developed to capture electronic momentum and model vibrational circular dichroism─and which we have postulated should also describe the Einstein–de Haas effect, a macroscopic manifestation of angular momentum conservation─is also able to recover the Λ-doubling energy splitting (or Λ-splitting) of the NO molecule nearly quantitatively and nonperturbatively (without a sum over states). The key observation is that, by parametrizing the electronic Hamiltonian in terms of both nuclear position (X) and nuclear momentum (P), a phase space method yields potential energy surfaces that explicitly include the electron-rotation coupling and correctly conserve angular momentum (which we show is essential to capture Λ-doubling). The data presented in this manuscript offer another small glimpse into the rich physics that one can learn from investigating phase space potential energy surfaces E_PS(X,P) as a function of both nuclear position and momentum, all at a computational cost comparable to standard Born–Oppenheimer electronic structure calculations.